Problem 24 - Algebra A Combined Function

Question

Write each as a logarithmic equation. $$ 10^{-2}=\frac{1}{100} $$

Step-by-Step Solution

Answer

\(-2 = \log_{10}\left(\frac{1}{100}\right)\)

Step 1: Identify the Components of the Exponential Equation

The given equation is \(10^{-2} = \frac{1}{100}\). This is in the form of \(a^b = c\), where \(a = 10\), \(b = -2\), and \(c = \frac{1}{100}\).

Step 2: Apply the Definition of Logarithms

Using the definition of logarithms, we can express an exponential equation \(a^b = c\) as a logarithm: \(b = \log_a(c)\). In this case, \(-2 = \log_{10}\left(\frac{1}{100}\right)\).

Step 3: Write the Logarithmic Equation

The logarithmic form of the given exponential equation is \(-2 = \log_{10}\left(\frac{1}{100}\right)\). This completes the transformation of the exponential equation into its logarithmic form.

Key Concepts

Exponential Equations

Exponential equations involve a mathematical expression in which a variable appears as an exponent. These equations are commonly written in the form \(a^b = c\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is the result of the expression. In the context of our problem, the exponential equation is \(10^{-2} = \frac{1}{100}\).
Exponential equations are significant because they allow us to solve for unknowns in situations where change happens rapidly or gradually, such as in growth models or decay processes in fields like finance or science.
An important aspect of understanding these equations is knowing how to manipulate them, often transforming them into logarithmic forms, especially when solving for the exponent or the base.

Logarithm Definition

A logarithm is a mathematical operation that helps us find the power to which a number, called the base, must be raised to obtain another number. Formally, if \(a^b = c\), then \(b = \log_a(c)\). In our problem, the expression \(-2 = \log_{10}(\frac{1}{100})\) shows the logarithm in action.
Logarithms play a vital role in mathematics, providing a way to deal with equations involving exponential growth or decay. Here are some vital points about logarithms:

• They allow for the transformation of multiplication into addition, making complex calculations easier.
• Logarithms are used in diverse fields, including science, engineering, and finance, for solving exponential equations efficiently.

Understanding the definition of logarithms will enable you to see how they are used to simplify and solve mathematical problems involving exponential relations.

Transformations in Mathematics

Transformations in mathematics refer to changing a mathematical expression or figure into a different form without altering its values or properties. One common transformation is converting exponential equations into logarithmic form. This conversion relies on the definition of logarithms. For example, transforming the exponential equation \(10^{-2} = \frac{1}{100}\) into its logarithmic form \(-2 = \log_{10}(\frac{1}{100})\).
Transformations are important as they:

• Provide different perspectives to solve complex problems.
• Offer methods to simplify equations and make calculations more manageable.

You commonly see them in algebra and calculus, where transformations aid in solving equations and understanding functions better.

Base of Logarithms

The base of a logarithm is the number that is raised to a particular power to achieve another number. It is a crucial component in understanding logarithmic equations. In our example, the base is 10, which is evident in \(-2 = \log_{10}(\frac{1}{100})\).
Different bases can give different types of logarithms:

• Base 10 is common in scientific calculations (common logarithms).
• Base \(e\) (approximately 2.718) is used for natural logarithms, prevalent in natural process descriptions.

Choosing the right base depends on the context and problem requirements. Understanding how bases work helps in seamlessly converting and interpreting between exponential and logarithmic forms, making solving and understanding equations more intuitive.